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Properties

Label 2-2888-2888.2667-c0-0-0
Degree $2$
Conductor $2888$
Sign $-0.380 - 0.924i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.926 − 0.376i)2-s + (−0.582 − 0.130i)3-s + (0.716 + 0.697i)4-s + (0.490 + 0.340i)6-s + (−0.401 − 0.915i)8-s + (−0.582 − 0.274i)9-s + (0.158 − 1.91i)11-s + (−0.326 − 0.499i)12-s + (0.0275 + 0.999i)16-s + (−1.26 + 1.22i)17-s + (0.435 + 0.473i)18-s + (−0.926 + 0.376i)19-s + (−0.868 + 1.71i)22-s + (0.114 + 0.585i)24-s + (−0.998 + 0.0550i)25-s + ⋯
L(s)  = 1  + (−0.926 − 0.376i)2-s + (−0.582 − 0.130i)3-s + (0.716 + 0.697i)4-s + (0.490 + 0.340i)6-s + (−0.401 − 0.915i)8-s + (−0.582 − 0.274i)9-s + (0.158 − 1.91i)11-s + (−0.326 − 0.499i)12-s + (0.0275 + 0.999i)16-s + (−1.26 + 1.22i)17-s + (0.435 + 0.473i)18-s + (−0.926 + 0.376i)19-s + (−0.868 + 1.71i)22-s + (0.114 + 0.585i)24-s + (−0.998 + 0.0550i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $-0.380 - 0.924i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ -0.380 - 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1032642680\)
\(L(\frac12)\) \(\approx\) \(0.1032642680\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.926 + 0.376i)T \)
19 \( 1 + (0.926 - 0.376i)T \)
good3 \( 1 + (0.582 + 0.130i)T + (0.904 + 0.426i)T^{2} \)
5 \( 1 + (0.998 - 0.0550i)T^{2} \)
7 \( 1 + (0.879 - 0.475i)T^{2} \)
11 \( 1 + (-0.158 + 1.91i)T + (-0.986 - 0.164i)T^{2} \)
13 \( 1 + (0.821 - 0.569i)T^{2} \)
17 \( 1 + (1.26 - 1.22i)T + (0.0275 - 0.999i)T^{2} \)
23 \( 1 + (-0.904 + 0.426i)T^{2} \)
29 \( 1 + (-0.851 - 0.523i)T^{2} \)
31 \( 1 + (-0.245 - 0.969i)T^{2} \)
37 \( 1 + (0.986 + 0.164i)T^{2} \)
41 \( 1 + (-0.450 - 0.392i)T + (0.137 + 0.990i)T^{2} \)
43 \( 1 + (-0.553 - 1.47i)T + (-0.754 + 0.656i)T^{2} \)
47 \( 1 + (-0.350 - 0.936i)T^{2} \)
53 \( 1 + (-0.350 - 0.936i)T^{2} \)
59 \( 1 + (0.207 + 0.180i)T + (0.137 + 0.990i)T^{2} \)
61 \( 1 + (0.298 - 0.954i)T^{2} \)
67 \( 1 + (-1.79 - 0.198i)T + (0.975 + 0.218i)T^{2} \)
71 \( 1 + (0.298 + 0.954i)T^{2} \)
73 \( 1 + (1.08 - 1.05i)T + (0.0275 - 0.999i)T^{2} \)
79 \( 1 + (0.754 - 0.656i)T^{2} \)
83 \( 1 + (1.49 - 0.810i)T + (0.546 - 0.837i)T^{2} \)
89 \( 1 + (1.26 + 1.22i)T + (0.0275 + 0.999i)T^{2} \)
97 \( 1 + (0.993 - 0.110i)T + (0.975 - 0.218i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.037109808614510977332341907772, −8.409565525948857202190750654692, −8.107862012505949118281082372415, −6.82543086769786220665832908223, −6.07579416717980179047461653281, −5.89259039447798151142173526604, −4.27185264862264288780175135732, −3.45883617622868774148631215919, −2.51909901200301542611988488978, −1.27554893676531578448338871922, 0.095301779150380037392641030300, 1.95788657970563179877133737649, 2.53324676071691155312333671054, 4.29544290849390198695623009256, 4.95332837872075151684461382100, 5.75679790287887329517257654992, 6.73660136431655100980082316600, 7.07655101346142999665397675841, 7.955397991970536567455077783060, 8.814771723797245457255825805292

Graph of the $Z$-function along the critical line